Related papers: On the distance between perfect numbers

Inspired by a recently formulated conjecture by Bannai et al. we investigate spherical codes which admit exactly three different distances and are spherical 5-designs. Computing and analyzing distance distributions we provide new proof of…

In this article we present set of infinite natural numbers which satisfies the conjecture $3n+1$.

We shall show that $9$ is the only odd infinitary superperfect numbers.

The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The…

Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values…

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of…

Two permutations of the natural numbers diverge if the absolute value of the difference of their elements in the same position goes to infinity. We show that there exists an infinite number of pairwise divergent permutations of the…

Let $k\ge2$ be an integer. A natural number $n$ is called $k$-perfect if $\sigma(n)=kn.$ For any integer $r\ge1$ we prove that the number of odd $k$-perfect numbers with at most $r$ distinct prime factors is bounded by $k4^{r^3}$.

We prove a refined version of Markov's theorem in Diophantine approximation. More precisely, we characterize completely the set of irrationals $x$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$ has only finitely many rational…

By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis $q$, we show that there are infinitely many pairs of…

For every $n = 2^k > 8$ there exist exactly $[(k+1)/2]$ mutually nonequivalent $Z_4$-linear extended perfect codes with distance 4. All these codes have different ranks.

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…

We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal…

Many generating series of combinatorially interesting numbers have the property that the sum of the terms of order $<p$ at some suitable point is congruent to a zero of a zeta-function modulo infinitely many primes $p$. Surprisingly, very…

A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if…

We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the from $y^2=x(x^2+b)$, where $b$ is any positive integer. We achieve this by using the…

Let $\mathcal{S}$ be a finite set of integer points in $\mathbb{R}^d$, which we assume has many symmetries, and let $P\in\mathbb{R}^d$ be a fixed point. We calculate the distances from $P$ to the points in $\mathcal{S}$ and compare the…

In this paper we study circles tangent to conics. We show there are generically $184$ complex circles tangent to three conics in the plane and we characterize the real discriminant of the corresponding polynomial system. We give an explicit…

The Erd\H{o}s-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter~$\delta$, at most $O(\delta^2)$ points…

We show that there are infinitely many triples of positive integers a, b, c (greater than 1) such that ab + 1, ac + 1, bc + 1 and abc + 1 are all perfect squares.